Substrate with a Structured Surface and Methods for the Production Thereof, and Methods for Determining the Wetting Properties Thereof

ABSTRACT

An implant includes a microstructured hyperhydrophilic surface with protrusions and depressions in which a spacing between the protrusions as a statistical mean is in a range of 1 to 100 μm and a profile height of the protrusions and depressions as a statistical mean is in the range of 1 to 80 μm.

This U.S. patent application is a Divisional of U.S. Ser. No. 14/364,312filed on 11 Jun. 2014, which is a national stage application ofPCT/DE2012/100382 filed on 16 Dec. 2012 and claims priority of Germanpatent document DE 10 2011 056 549.3 filed on 16 Dec. 2011, theentireties of which are incorporated herein by reference.

BACKGROUND OF THE INVENTION

The invention concerns a substrate with a structured surface and methodsfor the production of the substrate with a structured surface and alsomethods for determining the wetting properties of that substrate. Themethod according to the invention can be used to produce in particularsubstrates with user-specific fixed surface properties—implants arenamed by way of example—. Thus implants like for example dental implantsor endoprotheses which are distinguished by particularly good ingrowthat the implantation location in the jaw or extremity bones can thus beproduced with the method according to the invention.

In recent years it has become increasingly clear that the roughness ofthe surface of an implant, besides hydrophilicity and hydrophobicity ofthe implant surface, plays one of the most important parts in theintegration of implants. Roughness can increase by hydrophilicity andalso hydrophobicity. Thus it is known in the state of the art that anSLA (sand-blasted acid etched) surface exhibits a substantially betteringrowth behaviour than the smooth machined form of titanium. Besidesthe SLA surface with a roughness there are implants with a TPS (titaniumplasma sprayed) surface, which exhibits a roughness with a betterintegration healing behaviour.

The presence of a rough surface is always linked to an increase insurface area in comparison with a smooth surface. Thus for example theSLA and TPS surfaces, in comparison with a smooth surface, can have asurface area which is 2-20 times greater, and that has a positive effectin particular in the case of ingrowth in animals and humans.

A disadvantage of rough surfaces is the problem of removal in the eventof implant revisions. What is common in particular to the previouslyproduced implants is that the outwardly facing surfaces of the objectgenerally have irregular structures which adversely affect the ingrowthbehaviour in particular for use of the objects as implants, and do notpositively influence same. In addition titanium particles can becomedetached from the TPS surface and pass into the tissue.

Added to that is a reproduction capability, which needs to be improved,of the implants produced in that way as implants produced both using theSLA method and also using the TPS method exhibit a certain statisticalbreadth in respect of the surface properties and it is thereforenecessary to observe with the utmost accuracy the method parameters independence on the starting material for the purposes of standardisingthe implants.

Consideration was given on the part of the inventor to improving thesurface properties and it was discovered that an optimally structured ofthe implant can be afforded with a microstructure. It was demonstratedby the inventor that reverse engineering leads to a surface withproperties which are improved in relation to the two above-mentioned SLAand TPS surfaces, wherein the improved surfaces can be produced with alower risk potential.

It was further discovered on the part of the inventor that such roughimplant surfaces can be made further hyperhydrophilic, as is describedhereinafter, by means of wet-chemical methods and/or byfunctionalisation with hydrophilic organic molecules.

It will be noted however that the production of such hyperhydrophilicsurfaces generally requires the use of highly heated acids and thecorresponding plasma chambers. In relation to those hyperhydrophilicsurfaces hitherto the dynamic contact angles were measured withultrapure water in the form of the advancing angle (θ_(V)) and thereceding angle (θ_(R)) in accordance with the observations of theinventor with the value zero (θ_(V)/θ_(R)=0°/0°). In reality the contactangles are in the imaginary range.

SUMMARY OF THE INVENTION

The invention is therefore directed to substrates with a microstructuredsurface which, if desired, is superimposed by a second smallermicrostructure and/or by a nanostructure, as well as methods for theproduction thereof, which have the desired hyperhydrophilic surfaceproperties.

According to the invention a regular microstructure of that kind can beproduced by means of various methods. These include structure-removingmethods like also structure-building methods which respectively make useof acting upon the object, or of powder, with energy-rich radiation.

In accordance with the information in Mays (2007) “A new classificationof pore sizes. Studies in Surface Science and Catalysis, 160, 57-62” inrelation to pore sizes structures/roughnesses can be appropriatelyclassified as follows:

Nanostructures: 0.1-100 nm Microstructures: 0.1-100 μm Millistructures:0.1-100 mm

In that respect the range 0.1-0.99 can be referred to as“submicrostructure”.

According to the invention laser removal can be highly selectively usedas the structure-removing method in order to remove individual layersfrom the substrate without significant damage to the subjacent layers orthe substrate. The removed structures can be both in point or line formand also over a surface area.

According to the invention the following are to be named asstructure-building or layer-building methods for the production ofthree-dimensional objects like implants: rapid prototyping, rapidtooling, rapid manufacturing, laser sintering, laser microsintering andEBM.

According to the invention laser microsintering can be used as a furthermethod of producing microstructures. In that respect processing ofceramic powders in high quality is also possible.

A basic prerequisite for the methods is generally that the geometricaldata of the product are present three-dimensionally and can be processedas layer data. According to the invention, from the existing CAD data ofthe component, the data are converted into a data format, for example anSTL format, in order to structure the surface of a blank in specificallytargeted fashion by means of the above-mentioned methods or to be ableto build up the blank in structured form from powder.

The known apparatuses, including for rapid prototyping methods,respectively have such an STL interface serving to provide geometricalinformation from three-dimensional data models.

Thus the inventor developed a method with which surfaces provided withregular/periodically recurring microstructures can be produced by thesurface of the blank being acted upon with energy-rich radiation in oneor more patterns, which can be represented from a periodic functionconverted into an STL data set, wherein either a structure-buildingmethod is employed using particulate material like metal powder orceramic powder, or a structure-removing method is employed. In the caseof a structure-building method an amount of powder present on the blankcan be acted upon with the energy-rich radiation in one or more stepsand the pattern can be produced on the blank.

As a less complicated and expensive alternative in accordance with theinvention a structure-removing method can be employed, in which thedesired structure is produced by removal of surface material.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1D show REM images of typically wide-spread and successfulrough surfaces in dentistry and orthopaedics with FIG. 1A and FIG. 1BSLA surface (sand-blasted acid etched); FIG. 1C and FIG. 1D TPS surface(titanium plasma spray method) and the insert of FIG. 1C: transferfracture edge of the TPS surface.

FIGS. 2A-2D shows basic shapes of surface protrusions (profiles) as aside view, with FIG. 2A (sinusoidal profile), FIG. 2B (rectangularprofile), FIG. 2C (triangular profile), and FIG. 2D (sawtooth profile).

FIGS. 3A-3D show unit cells with the respective associated rectangularfunctions and some profiles (FIGS. 3A and 3B), arrangement patterns(FIG. 3C) as a cross-sectional view, and hollow cylinders or stellateprofiles (FIG. 3D).

FIG. 4 shows a combination of a first microroughness with a second microroughness.

FIGS. 5A-5B show determination of a static (FIG. 5A) and dynamic (FIG.5B) contact angle on a superhydrophobic unmodified TPS surface withultrapure water.

FIG. 6 shows determination of a dynamic contact angle on a surface afterchemically “switching over” from a superhydrophobic to ahyperhydrophobic condition.

FIG. 7 shows an illustration of the Wilhelmy functions in an undefinedregion of FIG. 6 as imaginary contact angles in dependence on a depth ofimmersion.

FIG. 8 shows a diagram for conversion of data from a trigonometricfunction.

DETAILED DESCRIPTION OF THE INVENTION

More precisely the present invention in an embodiment concerns a methodfor the production of a substrate with a regularly microstructuredsurface with protrusions and depressions, wherein the spacing betweenthe protrusions as a statistical mean is in the range of 1.0 to 100 μmand the profile height of the protrusions and depressions as thestatistical mean (Ra-value) is in the range of 1 to 80 μm, whichcomprises the steps:

-   -   a) providing a powder or a powder mixture of a sinterable        material powder on a blank;    -   b) applying a layer of the metal powder to the surface of the        blank; and    -   c) acting on the layer of the material powder with energy-rich        radiation in a pattern which can be represented from a periodic        function converted into an STL data set so that material powder        is sintered on at least a partial region of the surface of the        blank with the formation of at least a partial region of the        pattern.

In the method the blank can be produced from solid material orlayer-wise by way of a sintering method from a sinterable materialpowder.

If required movement or displacement of the blank in the axial orhorizontal direction and successive repetition of steps b) to d) can beeffected so that a further partial region of the pattern, that adjoinsthe first partial region of the pattern, can be sintered.

In an embodiment the method includes the successive repetition of stepsb) to c) until the surface is completely covered with the desiredpattern.

As mentioned above, beside the method of building up a pattern by way oflaser sintering or EBM, it is also possible to produce a desired surfacemicrostructure by laser ablation. Thus the invention also concerns amethod for the production of a substrate with a microstructured surface,which has the following steps:

-   -   a) providing a blank;    -   b) acting on the blank with energy-rich radiation at least        partially in a pattern which can be represented from a periodic        function converted into an STL data set so that the blank is        ablated with the formation of at least a partial region of the        pattern on at least a partial region of the surface.

The blank obtained having a regularly microstructured surface by thebuilding-up or removing method can be subjected to a treatment forproducing a second regular microstructure using a periodic functionconverted into an STL data set and/or a wet-chemical treatment forproducing a nanostructure.

If required the blank can be moved in the axial or horizontal directionand step b) can possibly be repeated a plurality of times until thesubstrate surface is provided at least in a partial region with thedesired microstructured pattern.

The invention thus also concerns a substrate in which thehyperhydrophilic surface is microstructured irregularly or at least in apartial region regularly.

The material of the blank can be selected from the group of metals,metal alloys, ceramic material (for example zirconium oxide), glassesand polymers (PEEK, polyether ether ketone) and combinations thereof.

In that respect the material of the blank in particular for use as animplant preferably comprises a material selected from the group ofmetals, metal alloys and combinations thereof with ceramic materials.Preferably the implant material used comprises metallic materials likepure titanium or metallic titanium alloys,chromium/nickel/aluminium/valadium/cobalt alloys (for example TiAlV4,TiAlFe2,5), high-quality steels (for example V2A, V4A, chromium nickel316L) or a combination thereof with ceramic materials likehydroxylapatite, zirconium oxide, aluminium oxide, in which the metallicis present as a composite material with ceramic material. Thenon-metallic materials including the polymers like PEEK can however alsobe used alone without combination with other materials.

According to the invention the microstructured surfaces obtained in thatway can be further hydrophilised by a wet-chemical treatment with forexample chromo-sulphuric acid, wherein the contact angle when wettingwith water can then no longer be measured in accordance with the classicmeasurement and evaluation procedure or is given as zero, but inaccordance with the novel method developed by the inventor can bespecified with imaginary numbers.

Such a treatment can be carried out for example in such a way that thesurface of the microstructured implant is treated with an oxidisingagent insofar as the preferably degreased implant is shock-heated in hotchromo-sulphuric acid—preferably in that respect the chromo-sulphuricacid is of a density of more than 1.40 g/cm³—at a temperature of above200° C., that is to say it is heated to the temperature of thechromo-sulphuric acid by immersion within a few seconds and is leftthere at that temperature for a period of 10 to up to 90 minutes,preferably up to 60 minutes, particularly up to 30 minutes, andthereafter directly after removal the implant is cooled down to ambienttemperature within a period of less than a minute, preferably within afew seconds. That can preferably be effected by the implant beingquenched by immersion in concentrated sulphuric acid at a temperature of15° C. to 25° C. In order to remove residues of acid and, if present,metal ions which are foreign to the implant, for example chromium ions,the surface of the metal implant is washed in a plurality of washingsteps (up to 15) with distilled water. If thereafter chromium ions arestill to be detected on the surface of the implant then the implant canbe treated with a solution of a complexing agent until no further metalions can be detected. The inventor surprisingly found that, when usingEDTA as the complexing agent the solution is coloured as brown-violetviolet when chromium is dissolved out of the samples. The inventorspropose accordingly for the situation that the samples are washed in 10%EDTA (1-3×) at pH 7, if required also in boiling EDTA solution, until nofurther colouration by chromium ions occurs.

Thus, by means of this method according to the invention, it is possibleto obtain an implant with a hyperhydrophilic surface, which can be madestorable in accordance with a further configuration of a method inaccordance with EP 2 121 058.

There the inventors carried out tests which gave surprising results incomparison with the teachings known in the state of the art. Because ofthe increased cost of wet packagings for preserving hydrophilic andultrahydrophilic surfaces on implants which, in the case of theultrahydrophilic metal implants according to the invention, surprisinglypermit storage-stable implants without a loss of wettability even atrelatively high levels of salt concentration of more than 0.5 M/l,liquid-free packaging methods were also sought. In that respect it wasfound that hyperhydrophilic surfaces on which salt solutions were leftto evaporate also became stable in relation to the loss of wettability.Evaporation can be effected under a protective gas or in atmosphericair, wherein the latter has been used as standard because of the aspectof simplicity.

After evaporation a fine macroscopically invisible “exsiccation layer”was formed on the surface treated in that way, which layer in accordancewith the invention stabilises and protects both the ultrahydrophilicityand also hyperhydrophilicity. In general, in accordance with theinvention, it is possible to use neutral salt solutions in solution of asingle salt or also various salts in a concentration and amount which isinert in relation to the ultrahydrophilic surface and is sufficient tocover the surface of the implant with the exsiccation layer afterevaporation. Evaporation can be performed when the implant is in thesolution of neutral salt, or when the implant has been removed from thesolution and is thus covered only with a thin layer of that solution. Acorresponding consideration also applies to the hyperhydrophilicimplants described here.

The invention therefore also includes a method which besides theabove-described steps for the production of the substrate, includes theadditional step that the surface obtained is protected, stabilised andmade capable of long-term storage of means of a solution of non-volatilesubstances like salts, organic solvents which do not interact with thesurface, or a salt-bearing exsiccation layer, to protect the surface ofthe substrate in relation to deterioration due to aging or sterilisationmethods (for example gamma sterilisation).

The invention is also directed to a substrate with a microstructuredhyperhydrophilic surface with protrusions and depressions, wherein thespacing between the protrusions as a statistical mean is in the range of1 to 100 μm and the profile height of the protrusions and depressions asa statistical mean (Ra value) is in the range of 1 to 80 μm, wherein atleast one of the two dynamic contact angles (θ_(V) and θ_(R)) is in thehyperhydrophilic range with ΔF/(P·γ)>1.0 to 2.15 (θ_(ai)>0.0i°−80i°), inparticular with ΔF/(P·γ)>1.0 to 1.0619 (θ_(ai)>0.0i°−20i°).

The invention further includes a substrate as defined hereinbefore inwhich the first microstructure with first protrusions and depressions issuperimposed by a second microstructure with second protrusions anddepressions, wherein the spacing between the second protrusions as astatistical mean is in the range of 0.1 to 10 μm and the height of thesecond protrusions and depressions as a statistical mean (Ra value) isin the range of 0.1 to 10 μm.

Preferably the spacing between the second protrusions as a statisticalmean is in the range of 0.1 to 5 μm and the height of the secondprotrusions is in the range of 0.1 to 5 μm.

The microstructure with the first protrusions and depressions or themicrostructure with the second protrusions and depressions can besuperimposed by a nanostructure which can be produced by a wet-chemicaltreatment, for example by acid etching, as described hereinafter.

It is advantageous if the surface has a regular first structure and thespacings and heights of the first protrusions are in the above-definedlimits. That first microstructure is preferably produced by the surfaceof the blank being acted upon with energy-rich radiation in a patternwhich can be represented from a periodic function converted into an STLdata set. That periodic function is preferably a trigonometric basicfunction A_(R)(x) which is selected from:

$\begin{matrix}{\mspace{20mu} {{A_{R}(x)} = \left( {{\sin (x)},} \right.}} & (1) \\{{{A_{R}(x)} = {\frac{4a}{\pi}\left( {{\sin (x)} + {\frac{1}{3}{\sin \left\lbrack {3x} \right\rbrack}} + {\frac{1}{5}{\sin \left( {5x} \right)}} + {\frac{1}{7}{\sin \left( {7x} \right)}} + {\frac{1}{9}{\sin \left( {9x} \right)}} + \ldots}\mspace{14mu} \right)}},} & (2) \\{{{A_{R}(x)} = {\frac{4a}{\pi}\left( {{\sin (x)} - {\left( \frac{1}{3} \right)^{2}{\sin \left( {3x} \right)}} + {\left( \frac{1}{5} \right)^{2}{\sin \left( {5x} \right)}} - {\left( \frac{1}{7} \right)^{2}{\sin \left( {7x} \right)}} + {\left( \frac{1}{9} \right)^{2}{\sin \left( {9x} \right)}} + \ldots}\mspace{14mu} \right)}},} & (3) \\{{{A_{R}(x)} = {\frac{2a}{\pi}\left( {{\sin (x)} - {\frac{1}{2}{\sin \left\lbrack {2x} \right\rbrack}} + {\frac{1}{3}{\sin \left( {3x} \right)}} - {\frac{1}{4}{\sin \left( {4x} \right)}} + {\frac{1}{5}{\sin \left( {5x} \right)}} + \ldots}\mspace{14mu} \right)}},} & (4)\end{matrix}$

or derivatives thereof.

An optional second microstructure which is superimposed on the firstmicrostructure is preferably produced by the surface of the blank beingacted upon with energy-rich radiation in a pattern which can berepresented from a periodic function converted into an STL data set.That periodic function can preferably be a trigonometric basic functionA_(R)(x) as specified hereinbefore, which with other variables leads toa lesser “wavelength” and “amplitudes” of the microstructure.

The inventive development on the part of the inventor is based on therealisation that as an important parameter the profilometric arithmeticmean value of the roughness (Ra value) gives information about thetopography of the surface.

For that purpose, in accordance with the considerations on the part ofthe inventor, a reference line is placed on a substrate surface in sucha way that the area of the peaks and valleys becomes equal. In that caseRa is defined as the arithmetic mean of the absolute deviations of theprofile heights upwardly and downwardly in μm. The following simplifiedequation describes the Ra value:

R _(a)=(z ₁ +z ₂ +z ₃ + . . . z _(n))/n [μm]  (5)

The absolute value of the profile height (positive or negative height onthe y-axis) related to the profile reference line is named z. L is adefined measurement length (window) along the x-axis. In idealised termssuch a surface profile corresponds to a regular sinusoidal oscillationwith the extreme values±z in deviation from the reference line (zeroline) (see FIG. 2A). Besides the Ra value a second topographic parameteris also defined in μm as the maximum profile height Ry (=total ofhighest profile peak and deepest profile valley). Finally, it is to benoted that surfaces with identical Ra values can have non-identicalsurface profiles.

A further roughness parameter is the dimensionless microscopic roughnessfactor r_(m):

$\begin{matrix}{r_{M} = {\frac{{actual}\mspace{14mu} {area}}{{geometrical}\mspace{14mu} {area}} = \frac{A^{\prime}}{A}}} & (6)\end{matrix}$

wherein A′ represents the measured increased surface area in comparisonwith the calculated geometrical surface area A. The microscopicroughness factor r which is generally captured by means of a laserscanning microscope (LSM) gives information about the microscopicincrease in size of the surface area due to the increasing roughness.

In cell cultures it has been found that regular structures without sharpedges on the biomaterial surface of cells are preferred. It has furtherbeen found that, in surfaces with narrow-neck pores, as can occur forexample in TPS surfaces, biofilms are formed, which can cause implantloosening. The aim therefore is to provide a surface without such pores.

The structuring of the substrates surface in the pattern which can berepresented from a periodic function converted into an STL data setimparts to the substrate properties which can be influenced by avariation in given parameters like for example roughness (Ra values),periodicity value, microscopic roughness factor r_(M), spacing betweenthe protrusions or maximum profile height Ry. According to the inventionsubstrates and methods for the production of substrates are preferred,which have or produce a roughness parameter Ra in the range of 1-250 μm,preferably between 1 and 80 μm and particularly preferably between 2 and30 μm. According to the invention substrates and methods for theproduction of substrates are preferred, which have or produce aperiodicity value n (λ/2) in the range between 1 and 100 μm, preferablybetween 10 and 60 μm and particularly preferably between 2-30 μm.According to the invention substrates and methods for the production ofsubstrates are preferred, which have or produce a microscopic roughnessfactor r_(M) in the range between 2 and 50. According to the inventionsubstrates and methods for the production of substrates are preferred,which have or produce a spacing between the protrusions as a statisticalmean in the range of 1 to 100 μm. According to the invention substratesand methods for the production of substrates are preferred, which haveor produce a maximum profile height Ry in a range of 2 to 500 μm.

The inventor developed the idea for the production of implants with ahomogeneous and defined roughness by way of consideration of theroughness as a sine curve. That is shown in FIG. 2A in which a surfaceprofile is described by means of a sine curve. The curve can bedescribed with a wavelength of λ=32 μm and with an amplitude (=Ra valueof 3.13λ/2.50 μm). The defined profile function applies in accordancewith equation 1 for the variation in the parameters λ and profileheight:

$\begin{matrix}{z = {{A_{R}(x)} = {\left( {\sin \; \frac{2\pi}{\lambda}x} \right)P}}} & (7)\end{matrix}$

wherein x is the independent variable, λ is the wavelength and P is theprofile height. For the defined profile function in FIG. 2A the equationis:

$\begin{matrix}{z = {{A_{R}(x)} = {\left( {\sin \; \frac{2\pi}{32}x} \right)50}}} & \left( {7a} \right)\end{matrix}$

In that way it is possible to represent all desired surface profiles byway of the basic equations 1-4.

The generalisation of that principle is shown in FIGS. 2B-D with theassociated trigonometric basic functions, where it is shown that,besides a sinusoidal profile, it is also possible to produce for examplea rectangular profile, a triangular profile and a sawtooth profile. Allprofiles which can be described with trigonometric functions or seriescan be produced by means of the method according to the invention. Withthat mathematical tool it is also possible to draft the parameterssought with Ra values in the range of 1-80 μm and r_(m) values in therange of 2-50 in a CAD system and determine them for manufacture. Thussuch surface structures can be produced into the micrometer range bymeans of selective electron beam melting (SEBM), selective laser meltingor selective laser-assisted manufacture.

FIG. 3 shows that the surface roughness can be described on a surfacewith two sinusoidal profiles (coordinates: X/Z and YZ). It is furthershown how a surface with solid-quadratic profiles can be constructed bymeans of rectangular functions. FIG. 3A shows a unit cell with awavelength of λ/2=32 μm, that is to say every 32 μm (wave peak) there isa rectangular profile which in the X- and Y-coordinate involves the samespacing of λ/2=32 μm (wave trough). The amplitude (Z-axis) for bothsinusoidal functions is the identical value of 80 μm (5λ/4), whichcorresponds to the Ra value. The entire surface can be divided into unitcells (3λ×3λ) of a size of 192×192 μm. A unit cell here has 9 profiles(FIG. 3A). It is also possible to combine a plurality of wavelengths,for example in the X-direction λ/2=32 and in the Y-direction λ/2=32 and16 μm alternately (FIG. 3b ). In that way rectangular profiles can nowbe placed in the same unit cell (192×192 μm) (FIG. 3B). In the case ofthe rectangular profiles the surfaces can be calculated as follows:

F _(Profile)=2(xy+xz+yz)−(xy)  (3)

wherein x, y and z are the specified coordinates and the base area (xy)on which the profile stands has to be subtracted. For areas shown inFIGS. 3A and B and calculated in Table 2 (wave crests) the followingthen applies:

F _(Profile)=2[(λ/2)(λ/2)+(λ/2)(5λ/4)+(λ/2)(5λ/4)]−[(λ/2)(λ/2)]  (4)

It is however then necessary to calculate the free areas including thewave troughs.

F _(Bottom area)=3(λ_(Y)/2×L _(EZ))+9×(λ_(X)/2)²  (5)

(with the same spacings) or

F _(Bottom area)=3(λ_(Y)/2×L _(EZ))+12(λ_(x1)×λ_(x2))  (6)

(with different spacings)

wherein L_(EZ) represents the length of the unit square. Furthersurface-relevant patterns and profiles which were not taken intoconsideration in terms of calculation are shown in FIGS. 3C and 3D.

As the calculations in Table 2 show it is possible in that way to veryeasily construct surfaces with Ra values and r_(m) values. Thus for thesurface A (λ/2=32 μm) in Table 2 (see also FIG. 3A) that gives a r_(m)value of 6.0 with a Ra value of 80 μm. If now the Ra value is reduced to35 μm then the r_(m) value falls to 3.2 (surface B). If the wavelengthis reduced to λ/2=8 μm with a Ra value of 80 μm then that actually givesa surface having a r_(m) value=22.6 (surface E). If now the Ra value isreduced to 35 μm that gives a r_(m) value=11.3, which is also stillconsiderable (surface F).

When using a plurality of different wavelengths substantially highersurface values (surfaces G and H) can be obtained (see FIG. 1B, 12profiles/unit cell). An attractive possible way of increasing thesurface size would be the use of hollow cylinders or stellate profiles(see FIG. 3D). To sum up Table 2 clearly shows that with thetrigonometric approach and parameters in the μm range it is possible toreach Ra values in the range of 2-80 μm and r_(m) values in the range ofr_(m)=3.2-22.6 μm.

Thus, using the available technologies like selective electron beammelting (SEBM), selective laser melting or laser-assisted manufacture(Lasergravur, rapid manufacturing) it is thus possible to produce thesurfaces of all biomaterial solid bodies in that way by removal orbuilding up from powders. In that respect a microstructure is possibleeven at a resolution below 10 μm. The invention shows thatcomputer-controlled production of such implant surfaces is possible.

Advantages of the method according to the invention are thus:

-   -   better compatibility due to homogeneity of the surface    -   large surface areas by way of area increases of 20-40 times    -   increase in surface capacity for proteins and pharmaceuticals by        20-40 times    -   increase in ultrahydrophilia    -   avoidance of infections    -   pharmaceuticals reservoir in hollow cylinder profiles    -   computer-aided manufacture (laser and electron beam technology)

According to the invention it is possible to describe all surfacestructures as trigonometric functions which consequently can be directlyrepresented and simulated as 3D vector graphics in AutoCAD. There aremany different variation options in regard to the configuration of thesurface according to the invention and the production ofunitarially-structured, regular and highly complex surfaces ismathematically pre-defined and can be applied to any surface of anycomponent.

Conversion of the data from the trigonometric function for example for arapid prototyping method and an RP apparatus is diagrammatically shownin FIG. 8.

In a further step peptides like bone growth factors can be immobilisedon the microstructures according to the invention optionallysuperimposed with second microstructures and/or nanostructures,covalently or by means of physisorptive or chemisorptive bonding,presumably on the basis of hydrophilic interactions on the implantmaterial. Adsorptive bonding is also possible after a covalentmodification of the surface with amino propyl triethoxy silane (APS)(Table 1). That makes it possible to form a chemotactically actingand/or biologically active implant surface, with covalent bonding aso-called juxtacrine, leading to accumulation, proliferation anddifferentiation of bone cells. It is possible in that way to provideso-called biologically active implants which with molecules liberatedfrom the surface, even at a distance of 500 to 1000 μm, exhibit achemotactic action on cells, in the case of BMPs on osteoblasts.

Preferably adequate loading of the hydrophilised metal surface isachieved by the peptides being applied in a physiological buffersolution in a concentration which is sufficient to achieve a loading ofmore than 200 ng/cm², preferably more than 500 ng/cm² and morepreferably more than 1000 ng/cm² of the peptide on the oxide surface ofthe metal implant.

In general that loading is achieved with a physiological buffer solutionof peptides in a concentration of more than 1 μg/ml, preferably morethan 200 μg/ml buffer solution.

According to the invention the peptides are biomolecules which areadvantageous for biocompatibility of the implant insofar as theycounteract possible rejection of the implant and/or promote ingrowth ofthe implant.

As mentioned above preferably proteins from the class of TGF proteins,in particular bone growth-promoting proteins from the class of bonegrowth factors “Bone Morphogenic Proteins” or the class of vasculargrowth factors like VEGF or angiotropin or also ubiquitin can be used aspeptides. The expression “Transforming Growth Factor” (TGF) is used todenote in particular the group (sub-group) of the (i) “TransformingGrowth Factors beta” (TGF-β) and the group (sub-group) of the (ii) BoneMorphogenetic Proteins (BMP). The latter are osteo-inductive proteinswhich simulate bone augmentation and bone healing insofar as they causeproliferation and differentiation of precursor cells to giveosteoblasts. In addition they promote the formation of alkaliphosphatases, hormone receptors, bone-specific substances like collagentype 1, osteocalcin, osteopontin, osteonectin, Bone Sialoprotein (BSP)and finally mineralisation.

Advantageously for immobilisation purposes it is possible to use aprotein of that class alone, in combination with further members of thatclass or also together with biomolecules like proteins of other classesor low-molecular hormones or also antibiotics to improve immune defence.In that respect those further molecules can also be immobilised on thesurface by way of bonds which can be cleaved in the physiologicalmedium.

The invention is described in greater detail by means of theaccompanying Figures in which:

FIG. 1 shows REM images of typically wide-spread and successful roughsurfaces in dentistry and orthopaedics with A and B SLA surface(sand-blasted acid etched); C and D TPS surface (titanium plasma spraymethod) and insert FIG. 10: transfer fracture edge of the TPS surface.The arrow points to the fusion gap between the TPS layer of puretitanium and the base material of titanium alloy (Ti-6Al-4V);

FIG. 2 shows basic shapes of surface protrusions (profiles) as a sideview. In this case options for the tip of the profile are round, flatand pointed with λ=32 μm; z=50 μm.

The trigonometric equations describing the profiles are specified underthe Figures. All possible forms of the roughness can be described by wayof such Fourier series.

Associated 3D basic shapes are: A. hyperboloid; B. cuboid; C. pyramids;D. asymmetric pyramids;

FIG. 3 shows unit cells with the respective associated rectangularfunctions and some profiles (A & B) and arrangement patterns (C) as across-sectional view with:

-   -   A. unit cell 192 μm×192 μm with 9 profiles (λx and λy=64 μm,        z=Ra=80 μm) (see Table 2, surface A)    -   B. unit cell 192 μm×192 μm with 12 profiles (λx 64 μm and λy=32        μm, z=Ra=80 μm); (see Table 2, surface G)    -   C. arrangement patterns for profiles    -   D. profiles with different surface area values

3D basic form A-C: cuboid;

FIG. 4 shows a combination of a first microroughness (λ=64 μm) with asecond microroughness (λ=7.1 μm). The surface area of the macroroughnessis increased by the factor of 2.25 (1.5×1.5) by the illustrated secondmicro roughness.

FIG. 5 shows determination of the static (A) and dynamic (B) contactangle on a superhydrophobic unmodified TPS surface with ultrapure water;

FIG. 6 shows determination of the dynamic contact angle on a surfaceafter chemically “switching over” from the superhydrophobic to thehyperhydrophobic condition;

FIG. 7 shows an illustration of the Wilhelmy functions in the undefinedregion of FIG. 6 as imaginary contact angles in dependence on the depthof immersion.

Without modification of the above-mentioned microstructure themicrostructured surface was further nanostructured in a wet-chemicalprocess and reacted with amino propyl triethoxy silane for theadsorption of BMP-2. For calculation of the monolayer coverage withBMP-2 a footprint of the BMP-2 of 20 am² (1 μg BMP-2˜4.6 cm²) was usedfor a monomolecular coverage of the surface. Under the given conditionsthe r_(m) value determined with BMP-2 is well in conformity with theLSM-determined r_(m) values. The adsorption values obtained are set outin Table 1.

TABLE 1 BMP-2 adsorption (APS-surface) ng/cm² pro pro r_(M) geometr.actual r′_(M) (A′/A) area area (A′/A) Surface Ra μm LSM (A) (A′) BMP-2SLA ~2-3 2.5 394 ± 66 (6) 157 ± 27 (6) 1.8 TPS 30.0 ± 4.4 (4) 20 5221 ±293 (6) 261 ± 15 (6) 23.9Data format: ξ±S.D.

According to the invention the surface microstructures can be producedas desired using the trigonometric functions. For that purpose theroughness parameters can be used for a rectangular profile having aprofile height of 35 and 80 μm with the values as shown in Table 2.

TABLE 2 X Y λ_(x) λ_(y) λ/2 λ/2 F_(Profile) Profile/ Ra Surface μm μm μmμm μm² IC μm Ry r_(m) A 64 64 32 32 21504  9 80 160 6.0 B 64 64 32 32 9984 9 35 70 3.0 C 32 32 16 16 10 496  36 80 160 11.6 D 32 32 16 16 4 26636 35 70 6.9 E 16 16 8 8 5 248 144 80 160 22.6 F 16 16 8 8 2 304 144 3570 10.3 λ/2 + λ/4 G 64 32 32 32 + 16 21 504  12 80 160 7.8 H 16 8 8 8 +4 2 304 192 35 70 13.6

The surfaces A and G are shown in FIGS. 3A and 3B. The area of theindividual cell (IC) is 36 864 μm².

Thus, by means of the method according to the invention, it is possibleto provide substrates like implants with defined surface structureswhich are hyperhydrophilic directly after laser-technology production.If the surface is not sufficiently hyperhydrophilic it can be furtherhyperhydrophilised by means of a chemical hydrophilisation method. Thusthese surface structures which can also bear implants lead to particularwetting properties which according to the invention are identified ashyperhydrophilic surfaces. Such chemical hydrophilisation methodsinclude wet-chemical methods like acid etchings and also structuredsurfaces functionalised by covalent or non-covalent bonding of highlyhydrophilic molecules like polyethyene glycol (PEG),poly(2,3-dihydroxypropyl methacrylate) (PDHMA) orpoly[2-(methacryloyloxy) ethyl phosphorylcholine] (PMPC), wherein PDHMAand PMPC have a zwitterionic structure. Covalent coupling can beeffected for example by way of suitable triethoxy silane derivatives ofPEG, PDHMA and PMPC.

As is known in the state of the art, in the present application tocharacterise wettability hydrophilic surfaces with dynamic contactangles of a value of 0<θ<10° are referred to as ultrahydrophilic whilesurface with the determinable contact angles according to the inventionin the form of imaginary contact angles of a value of θ<0i to 1.4i radare referred to as hyperhydrophilic.

Usually measurement of the hydrophilicity properties of a surface iseffected on the basis of determining the contact angle. Introduction ofthe contact angle θ by Thomas Young more than 200 years ago basicallyopened the way in that respect to understanding wettability byintroduction of the Young equation:

γ_(sv)=γ_(sl)+γ_(lv) cos θ_(o)  (7)

wherein γ_(sv), γ_(sl) and γ_(lv) represent the surface tensions of thephase limits which are in contact of liquid (l), solid body (s) andvapour/gas phase (v), with θ₀ as the equilibrium contact angle forexample of a sitting drop. The Young equation which applies to acompletely smooth surface is not easy to solve as generally only γ_(lv)and θ₀ can be measured.

It is to be noted that the Young equation applies for contact anglesapplies for ideal, smooth, impermeable surfaces in thermodynamicequilibrium. Contact angles on real rough surfaces in contrast arereferred to with the attribute of “apparent” to distinguish them fromthe Young contact angle on an ideal surface.

Starting therefrom Ludwig Wilhelmy (Ann. Phys., 119, 177-217) foundabout 60 years later the Wilhelmy balance in which he linked tensiometryto contact angle measurements. In the case of force measurements bymeans of the Wilhelmy balance the sample is immersed in and removed fromultrapure water, with measurement of the force. The contact angle isthen calculated from the force of immersing and removing the sample inaccordance with the known Wilhelmy equation:

F=Pγ cos θ−Vgρ [N]  (8)

wherein F represents the measured nett force and in the first term onthe right-hand side of the equation P is the perimeter of the sample, γthe surface tension of the water and θ the dynamic contact angle(advancing angle θ_(V) or receding angle θ_(R)). In the second term Vdenotes the volume of the displaced liquid, g denotes gravity and ρdenotes the density of the liquid. The second term which specifies thebuoyancy of the sample in the liquid can be eliminated by extrapolationto the depth of immersion zero and leads to the simplified form of theWilhelmy equation:

cos θ=F/(P×γ)  (9)

If P is equated to the unit of 1 cm (for example plates of 10×5×1 mm)that gives the constant 1/(P·γ)=K_(θ)=1.39·10³ N⁻¹ and thus theequation:

cos θ=K _(θ) ·F  (9a)

If contact angles are calculated in accordance with Equation 9 withoutextrapolation to zero they are referred as “virtual” contact angles. Theuse of virtual contact angles is shown hereinafter in FIG. 7.

Considered practically the validity of Equation 9 for a completelysmooth surface is limited by two prohibited contact angles: (i) θ≯119°and (ii) θ≮0°. For the first case on the hydrophobic side it is knownfrom the state of the art that contact angles for physical reasonscannot exceed the value θ=110° C. In the second case (θ≮0°) on thehydrophilic side the contact angle cannot be less than zero for themathematical reason that cos θ>1 is not defined. It was now found on thepart of the inventor that this latter barrier in accordance with classicmathematical understanding can be overcome if the contact angles areexpanded into the imaginary numerical range ai and thus the surfaceproperties of the implants can be assessed.

For the measurement operations, the inventor used metal platescomprising a titanium alloy coated with pure titanium (Ti-6Al-4V) (socalled titanium plasma spray method TPS) with a roughness of Ra=30 μmand a microscopic roughness of r_(m)=20. Dynamic contact angles θ_(V)(advancing angle) and θ_(R) (receding angle) were determined withultrapure water using the Wilhelmy method (tensiometer DCAT 11,Dataphysics, Filderstadt, Germany). Immersion and removal speeds were 1mm/min (17 μm/s) so that the measured contact angles are independent ofthe dip speed. The apparent static contact angles θ_(S)′ (sitting dropmethod; 3-5 μl of ultrapure water) were graphically evaluated. Theimaginary contact angles were then calculated by the inventor from themeasured force values.

The established “extreme hydrophilia (θ>0i) (“hyperhydrophilia”)—here inthe absence of hysteresis on a microrough surface” after a wet-chemicaltreatment (acid etching)—was referred to by the inventor here by theterm “inverse Lotus effect”. That term is also used in the descriptionof hyperhydrophilic surfaces which occurred by way of so-called“chemical switching” from a surface exhibiting the “Lotus effect”. Thusa hyperhydrophobic surface (FIG. 5) is “switched over” into ahyperhydrophilic surface by treatment with chromo-sulphuric acid,wherein the latter in accordance with previous analytical methods has anextreme spreading of water (θ_(S) ^(H) ² ^(O)=0°) and dynamic contactangles of θ_(V)/θ_(R)=0°/0°) (FIG. 6). As n-hexane and mineral oilspread on those surfaces (θ_(S) ^(H) ² ^(O)/θ_(S) ^(Oil)/θ_(S)^(n-Hexane)˜0°/0°/0°), they are also referred to as superamphiphilic.The “reverse transformation” from the hyperhydrophilic condition intothe hydrophobic condition occurs spontaneous slowly in air if thesurface is not conserved.

The invention is also directed to a method in which the treatment toproduce a nanostructure includes the step of a wet-chemical treatment ofthe microstructured surface, wherein a hydrophobic or weakly hydrophilicsurface is converted into an ultrahydrophilic or hyperhydrophilicsurface, wherein at least one of the two dynamic contact angles (θ_(V)and θ_(R)) is in the range

ΔF/(P·γ)=0.980 to 2.15 and

preferably in the hyperhydrophilic range

ΔF/(P·γ)>1.0 to 1.0619 (θ_(a) ^(i)>0°−0.35i rad).

The observations on the part of the inventor relating to the sequentialoccurrence of two different Lotus effects on one and the same surfaceafter “chemical switching” indicates that there is a link between thosetwo effects, which however is still unclear. In the hydrophobic case theinfluence of roughness on the dynamic contact angle(θ_(V)′/θ_(R)′=98.8°/36.7°) by way of an increase to θ_(S) ^(H) ²^(O)˜145° (static method) by heterogeneous wetting is to be clearlyseen. On the hydrophilic side however there is lacking a similar effectin respect of surface roughness on a contact angle of zero. Themeasurements by the inventor showed that all contact angles which lay inthe region of cos θ>1 were outputted as contact angles of the valuezero. An evaluation according to the invention of the raw data of FIG. 6now shows that 17% of the measurement points in FIG. 6 give undefinedcontact angles with cos θ>1. That observation is illustrated in theprofile in FIG. 6 by a line of demarcation separating the defined fromthe undefined region. The inventor now found a way of bringing the dataof the Wilhelmy measurements from the undefined condition into a definedcondition.

TABLE 3 Continuous range of the imaginary and real contact anglesValidity range arccos (K_(θ) · F), rad θ_(ai) ^(a), degrees K_(θ) · F1.40i  80.21i 2.1509 1.23i  70.47i 1.8568 1.05i  60.16i 1.6038 0.87i 49.85i 1.4029 0.71i  40.68i 1.2628 0.53i  30.37i 1.1438 Central region0.35i  20.05i 1.0619 of the inverse 0.18i  10.31i 1.0162 Lotus effect0.04i  2.29i 1.0008 0.00  0i +1.0 Hyperhydrophilia

cos θ ≥ 1) Ultrahydrophilia cos θ ≤ 1)

0.00  0 +1.0 0.04  2.29 0.9992 0.18  10.31 0.9838 0.35  20.05 0.93970.53  30.37 0.8628 0.71  40.68 0.7584 0.87  49.85 0.6448 1.05  60.160.4976 1.23  70.47 0.3342 1.40  80.21 0.1699 1.57  90.00 0 3.14 180.00−1.0 K_(θ)

-   (i) Table 3 shows classic and novel imaginary contact angles θ_(ai)    ^(a) in radians and degrees, which were calculated in accordance    with the (K_(θ)·F) values. In that respect imaginary and real    contact angle series behave like mirror images in relation to zero.    The “inverse Lotus effect” extends from 0.18 rad (˜10°) in the real    to 1.4i rad (˜80°) in the imaginary range of numbers preferably from    0.18 rad to 0.35 rad (˜20°). In that respect one of the dynamic    contact angles (for example θ_(A)) can be classic and the second    (for example θ_(ai,R)) can be imaginary, which is referred to as a    hybrid contact angle pair. On the other hand both dynamic contact    angles (θ_(ai,V)/θ_(ai,R)) can also be imaginary (pure imaginary    contact angle pair). Multiplication of the radian value by 180/π    leads to the contact angle in degrees: 57.3×0.4i [rad]=22.9i°. For    Kθ·F>1.0 there is an imaginary contact angle of >0.0i rad and >0.0i    degrees respectively. That is defined as the lower limit for    imaginary contact angles.

These findings give the expansion of the Wilhelmy equation into theimaginary range of numbers:

cos θ_(ai) ^(a) =F/(P×γ)  (10)

The general expression θ_(ai) ^(a) denotes all contact angles in thereal range (superscript a) for the boundary condition (Kθ·F)<1 and allcontact angles in the imaginary range (subscript ai) for the boundarycondition K_(θ)·F)>1 (see Table 3).

Now, by means of equation 10, it is possible to specify defined contactangles for all force measurements in the range (K_(θ)·F)=1.0 to +2.15starting from the real number system of cos(180°) to the imaginarysystem of cos(80i°) (see Table 3). Larger imaginary contact angles up to180i° are conceivable on rough surfaces according to the inventor'sassumptions.

The use of imaginary contact angles for determining a highly wettableTPS surface is shown in FIG. 7. 45 representative values of the raw dataabove the line of demarcation at 0.102 g in FIG. 6 were selected andtheir force values (K_(θ)·F; in the range 1.00 to 1.07) were convertedinto virtual imaginary contact angles (θai) and plotted as a function ofthe depth of immersion. Extrapolation of the linear component of thecurves to the position zero of the depth of immersion gave the apparentimaginary advancing and receding angles(θ_(ai,V)/θ_(ai,R)=0.36i°/0.37i°. The imaginary contact anglesascertained in that way are a complex function of the four wettingparameters cohesion, adhesion, spread and immersion. They contain theitems of information in respect of those wetting parameters includingwater absorption and are therefore characteristic of wetting of theillustrated rough surface.

Thus, on the basis of the inventor's realisations, it is possible todetermine the properties of such hydrophilic surfaces, for whichhitherto such determinations were not possible as in the case ofhyperhydrophilic surfaces, and to be able to assess their suitabilityfor subsequent treatments including coating operations.

Thus the invention is also directed to a method for determining thewetting properties of the surface of a substrate, which includes thesteps:

-   -   a. carrying out a Wilhelmy/force measurement for ascertaining        (K_(θ)·F);    -   b. calculating the apparent contact angles θ_(V) and θ_(R) on        the basis of the result of step a), wherein said calculation        -   i. is effected for the situation where (K_(θ)·F)≤1 in            accordance with arccos (K_(θ)·F)=real contact angles; and        -   ii. is effected for the situation where (K_(θ)·F)>1 in            accordance with arccos (K_(θ)·F)=imaginary contact angles;            and    -   c. determining the wetting properties of the substrate on the        basis of the contact angles θ_(V) and θ_(R) calculated in step        b).

The method according to the invention thus makes it possible tospecifically distinguish hyperhydrophilic surfaces from the hydrophilicsurfaces and sort out such materials. Thus the wetting properties can beclassified from hydrophobic by way of hydrophilic and superhydrophilicto ultrahydrophilic with the contact angles linked thereto.

The invention also concerns an apparatus for carrying out theabove-mentioned method which includes a measuring unit, an evaluationunit and an output unit, wherein the measuring unit is adapted for forcemeasurement of the Wilhelmy/force measurement, the evaluation unit isadapted to convert the measurement values obtained by the measuring unitby means an algorithm into an imaginary advancing angle (θ_(V)) andreceding angle (θ_(R)) and the output unit is adapted to further processthe contact angle obtained by the evaluation unit. In that respectfurther processing can include contact angle display in degrees orradians.

Thus it is possible on the basis of the inventor's realisations todetermine the properties of such hydrophilic surfaces, for whichhitherto such determination operations were not possible as in the caseof hyperhydrophilic surfaces and to be able to assess their suitabilityfor subsequent treatments including coating operations.

1. A method for the production of an implant with a regularlymicrostructured surface with protrusions and depressions, wherein thespacing between the protrusions as the statistical mean is in a range of1 to 100 μm and a profile height of the protrusions and depressions as astatistical mean is in a range of 1 to 80 μm, and wherein the implanthas a microscopic roughness factor r_(M) in a range between 2 and 50,comprising: a) providing a powder or a powder mixture of a sinterablematerial powder on a blank; b) applying a layer of the metal powder tothe surface of the blank; and c) acting on the layer of the materialpowder with energy-rich radiation in a pattern which can be representedfrom a periodic function converted into an STL data set so that materialpowder is sintered on at least a partial region of the surface of theblank with the formation of at least a partial region of the pattern. 2.A method according to claim 1 wherein the blank is produced from solidmaterial or layer-wise by way of a sintering method from a sinterablematerial powder.
 3. A method according to claim 1 wherein the blankobtained in c) with a regularly microstructured surface is subjected toa treatment for producing a second regular microstructure using aperiodic function converted into an STL data set and/or a wet-chemicaltreatment for producing a nanostructure.
 4. A method for the productionof an implant with a regularly microstructured surface, comprising: a)providing a blank; and b) acting on the blank with energy-rich radiationat least partially in a pattern which can be represented from a periodicfunction converted into an STL data set so that the blank is ablatedwith the formation of at least a partial region of the pattern on atleast a partial region of the surface.
 5. A method according to claim 4in which the blank obtained in b) with a regularly microstructuredsurface is subjected to a treatment for producing a second regularmicrostructure using a periodic function converted into an STL data setand/or a wet-chemical treatment for producing a nanostructure.
 6. Amethod according to claim 3 wherein the treatment for producing ananostructure includes the step of a wet-chemical treatment of themicrostructured surface, wherein a hydrophobic or weakly hydrophilicsurface is converted to an ultrahydrophilic or hyperhydrophilic surface,wherein at least one of the two dynamic contact angles θ_(V) and θ_(R)is in the hyperhydrophilic range with 1.0<ΔF/P·γ≤2.15, whereinθ_(ai)>0.0i°−80i°, wherein θ_(V) stands for advancing angle, θ_(R)stands for receding angle, ΔF stands for a difference between measurednet forces, γ stands for surface tension of water, and P stands forperimeter of the sample.
 7. A method according to claim 6 which furthercomprises the additional step that the surface obtained is protected,stabilised and rendered capable of long-term storage by a solution ofnon-volatile substances like salts, organic solvents which do notinteract with the surface, or a salt-bearing exsiccation layer forprotecting the surface of the substrate in relation to a reduction inwetting with a loss of hyperhydrophilia due to aging or stabilisationmethods.
 8. A method according to claim 1 in which the periodic functionconverted into an STL data set is a trigonometric function A_(R)(x)selected from the group consisting of: $\begin{matrix}{\mspace{20mu} {{A_{R}(x)} = \left( {{\sin (x)},} \right.}} \\{{{A_{R}(x)} = {\frac{4a}{\pi}\left( {{\sin (x)} + {\frac{1}{3}{\sin \left\lbrack {3x} \right\rbrack}} + {\frac{1}{5}{\sin \left( {5x} \right)}} + {\frac{1}{7}{\sin \left( {7x} \right)}} + {\frac{1}{9}{\sin \left( {9x} \right)}} + \ldots}\mspace{14mu} \right)}},} \\{{{A_{R}(x)} = {\frac{4a}{\pi}\left( {{\sin (x)} - {\left( \frac{1}{3} \right)^{2}{\sin \left( {3x} \right)}} + {\left( \frac{1}{5} \right)^{2}{\sin \left( {5x} \right)}} - {\left( \frac{1}{7} \right)^{2}{\sin \left( {7x} \right)}} + {\left( \frac{1}{9} \right)^{2}{\sin \left( {9x} \right)}} + \ldots}\mspace{14mu} \right)}},} \\{{{A_{R}(x)} = {\frac{2a}{\pi}\left( {{\sin (x)} - {\frac{1}{2}{\sin \left\lbrack {2x} \right\rbrack}} + {\frac{1}{3}{\sin \left( {3x} \right)}} - {\frac{1}{4}{\sin \left( {4x} \right)}} + {\frac{1}{5}{\sin \left( {5x} \right)}} + \ldots}\mspace{14mu} \right)}},}\end{matrix}$ and derivatives thereof.
 9. A method according to claim 1in which the roughness parameter is in the range of between 1 and 80 μm.10. A method according to claim 1 in which a periodicity value n(λ/2) isin the range of between 1 and 100 μm.
 11. A method for determining thewetting properties of the surface of a substrate, comprising: a)carrying out a Wilhelmy/force measurement for ascertaining (K_(θ)·F), b)calculating the apparent contact angles θ_(V) and θ_(R) on the basis ofthe result of step a), wherein said calculation i. is effected for thesituation where (K_(θ)·F)≤1 in accordance with arccos (K_(θ)·F)=realcontact angles; and ii. is effected for the situation where (K_(θ)·F)>1in accordance with arccos(K_(θ)·F)=imaginary contact angles; and c)determining the wetting properties of the substrate on the basis of thecontact angles θ_(V) and θ_(R) calculated in step b).
 12. An apparatusfor carrying out the method according to claim 11 comprising: ameasuring unit, an evaluation unit and an output unit, wherein themeasuring unit is adapted for force measurement of the Wilhelmy/forcemeasurement, wherein the evaluation unit is adapted to convert themeasurement values obtained by the measuring unit by an algorithm intoan imaginary advancing angle (θ_(V)) and receding angle (θ_(R)) andwherein the output unit is adapted to further process the contact angleobtained by the evaluation unit.
 13. The method according to claim 3,wherein at least one of two dynamic contact angles θ_(V) and θ_(R) is ina hyperhydrophilic range with 1.0<ΔF/P·γ≤1.0619, whereinθ_(ai)>0.0i°−20i°, wherein θ_(V) stands for advancing angle, θ_(R)stands for receding angle, ΔF stands for a difference between measurednet forces, γ stands for surface tension of water, and P stands forperimeter of the sample.